| L(s) = 1 | + 5.49e3·2-s − 1.52e6·3-s − 3.39e6·4-s + 1.00e9·5-s − 8.39e9·6-s − 2.02e11·8-s + 1.48e12·9-s + 5.54e12·10-s + 1.39e13·11-s + 5.17e12·12-s + 6.33e13·13-s − 1.54e15·15-s − 1.00e15·16-s + 6.78e14·17-s + 8.16e15·18-s + 1.52e16·19-s − 3.42e15·20-s + 7.65e16·22-s + 7.85e16·23-s + 3.10e17·24-s + 7.21e17·25-s + 3.48e17·26-s − 9.77e17·27-s − 1.15e18·29-s − 8.47e18·30-s + 8.59e17·31-s + 1.31e18·32-s + ⋯ |
| L(s) = 1 | + 0.948·2-s − 1.65·3-s − 0.101·4-s + 1.84·5-s − 1.57·6-s − 1.04·8-s + 1.75·9-s + 1.75·10-s + 1.33·11-s + 0.167·12-s + 0.754·13-s − 3.07·15-s − 0.888·16-s + 0.282·17-s + 1.66·18-s + 1.58·19-s − 0.186·20-s + 1.26·22-s + 0.747·23-s + 1.73·24-s + 2.42·25-s + 0.715·26-s − 1.25·27-s − 0.604·29-s − 2.91·30-s + 0.195·31-s + 0.201·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.543148442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543148442\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 5.49e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.52e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 1.00e9T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.39e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 6.33e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 6.78e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.52e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 7.85e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.15e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 8.59e17T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.89e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.33e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.57e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 5.03e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 5.46e20T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.15e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 4.67e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 8.62e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.69e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.82e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.19e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 9.97e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.70e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 2.07e24T + 4.66e49T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18909499669693638065277402658, −9.863994420550088166579497716384, −9.182093459190135789059384178719, −6.71960957485334599830012238226, −6.04663024184770920881783086319, −5.46392142134260543509878777979, −4.62437485918770600687302927375, −3.22363548810866443689162191542, −1.51267991150392141665236958075, −0.845999361203040906433918386758,
0.845999361203040906433918386758, 1.51267991150392141665236958075, 3.22363548810866443689162191542, 4.62437485918770600687302927375, 5.46392142134260543509878777979, 6.04663024184770920881783086319, 6.71960957485334599830012238226, 9.182093459190135789059384178719, 9.863994420550088166579497716384, 11.18909499669693638065277402658
